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 A054423 Number of unlabeled 3-gonal cacti having n triangles. 5
 1, 1, 1, 2, 7, 19, 86, 372, 1825, 9143, 47801, 254990, 1391302, 7713642, 43401974, 247216934, 1423531255, 8275108733, 48511773461, 286542497274, 1704002332513, 10195435737315, 61341136938138, 370933387552634, 2253475545208390, 13748639775492766, 84211761819147696 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also, the number of noncrossing partitions up to rotation composed of n blocks of size 3. - Andrew Howroyd, May 04 2018 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..200 Miklos Bona, Michel Bousquet, Gilbert Labelle, Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56 (pdf, dvi). M. Bousquet and C. Lamathe, Enumeration of solid trees according to edge number and edge degree distribution, Discr. Math., 298 (2005), 115-141. Index entries for sequences related to cacti FORMULA a(n) = ((Sum_{d|n} phi(n/d)*binomial(3*d, d)) + (Sum_{d|gcd(n-1, 3)} phi(d)*binomial(3*n/d, (n-1)/d)))/(3*n) - binomial(3*n, n)/(2*n+1) for n > 0. - Andrew Howroyd, May 04 2018 a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 2)). - Vaclav Kotesovec, Jun 01 2022 MAPLE with(combinat): with(numtheory): m := 3: for p from 1 to 40 do s1 := 0: s2 := 0: for d from 1 to p do if p mod d = 0 then s1 := s1+phi(p/d)*binomial(m*d, d) fi: od: for d from 1 to p-1 do if gcd(m, p-1) mod d = 0 then s2 := s2+phi(d)*binomial((p*m)/d, (p-1)/d) fi: od: printf(`%d, `, (s1+s2)/(m*p)-binomial(m*p, p)/(p*(m-1)+1)) od: # James A. Sellers, Mar 17 2000 MATHEMATICA a[0] = 1; a[n_] := (DivisorSum[n, EulerPhi[n/#] Binomial[3 #, #]&] + DivisorSum[GCD[n - 1, 3], EulerPhi[#] Binomial[3n/#, (n-1)/#]&])/(3n) - Binomial[3n, n]/ (2n + 1); Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *) PROG (PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(3*d, d)) + sumdiv(gcd(n-1, 3), d, eulerphi(d)*binomial(3*n/d, (n-1)/d)))/(3*n) - binomial(3*n, n)/(2*n+1))} \\ Andrew Howroyd, May 04 2018 CROSSREFS Column k=3 of A303694. Cf. A052393, A054422, A082938. Sequence in context: A362097 A080873 A126162 * A137990 A056650 A182169 Adjacent sequences: A054420 A054421 A054422 * A054424 A054425 A054426 KEYWORD nonn AUTHOR Simon Plouffe, Mar 15 2000 EXTENSIONS More terms from James A. Sellers, Mar 17 2000 Terms a(24) and beyond from Andrew Howroyd, May 04 2018 STATUS approved

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Last modified November 28 21:44 EST 2023. Contains 367419 sequences. (Running on oeis4.)